DIELECTRIC MEASUREMENT FOR SOLID CYLINDRICAL SAMPLES

Transcription

1 SCA24-2 1/12 DIELECTRIC MEASUREMENT FOR SOLID CYLINDRICAL SAMPLES N. Seleznev, A. Boyd, T. Habashy, C. Straley Schlumberger-Doll Research, Rdgefeld, US S. Luth, Delft Unversty of Technology, The Netherlands Ths paper was prepared for presentaton at the Internatonal Symposum of the Socety of Core Analysts held n Abu Dhab, UAE, 5-9 October, 24 ABSTRACT In ths paper we descrbe a coaxal-crcular cell for wde band delectrc measurements on sold cylndrcal samples. Combned wth exstng data-processng algorthms, the technque can be appled to a varety of materals ncludng olfeld rocs. The smple cylndrcal sample geometry s a dstnctve feature of ths method and s a sgnfcant advantage of the technque. Commonly used coaxal cells requre cylndrcal samples modfed wth a central hole to accommodate the nner electrode of the coaxal lne. Although ths confguraton presents no problem for lqud samples, for sold samples, precse machnng s requred to nsure good measurements. For roc materals, whch must be ground nto shape usng crcular damond cutters, creatng good coaxal samples for coaxal cells s an art; t can be dffcult, or mpossble, n case of wealy consoldated rocs. In addton, n a coaxal sample t s dffcult to acheve homogeneous partal saturaton. In contrast partal saturaton of sold cylndrcal samples can be acheved relatvely easy. Hence the capablty to measure smple cylndrcal samples that ths new cell provdes not only smplfes the sample preparaton but also allows for the partal saturaton experments. We dscuss the expermental set up, forward model and the nverson methodology, whch are requred for ths geometry. We also assess the uncertanty assocated wth the measurement. The forward model was valdated by the numercal modelng of the cell response. We obtaned a good ft between measured and expected results for several materals wth nown permttvty and conductvty ncludng brnes and glycerol. Permttvty and conductvty of brnes were calculated from the measurements of DC conductvty and temperature usng the Klen-Swft model [3]. Glycerol permttvty was taen from Bucley and Maryott [4]. Measurements of carbonate rocs at partal saturatons are presented. INTRODUCTION The nowledge of the permttvty and ts frequency behavor s mportant n both basc and appled research. Delectrc measurements are a partcularly nformatve technque for geophyscal applcatons [1]. In the laboratory delectrc propertes can be measured by dfferent methods employng varous sample szes and shapes [5]. Untl now coaxal cells were commonly utlzed [5-8]. The nowledge of both reflecton and transmsson coeffcents allows for closed form expresson dervaton and smplfes the computaton

2 SCA24-2 2/12 of permttvty and conductvty from expermental data. However these cells requre a coaxal sample. In some cases, creatng a coaxal sample s easy, as wth a lqud sample. In other cases the nabllty to precsely shape the sample can lmt the use of a coaxal cell. In case of wealy consoldated materals t s dffcult to machne such a sample. Ths s especally true for roc materals whch must be ground nto shape rather than usng more conventonal machnng methods. In addton to performng delectrc measurements, we would le to do other partal saturaton measurements such as low frequency resstvty and NMR. The coaxal geometry s not sutable for partal saturaton experments n the case of porous samples and the ancllary measurements requre cylndrcal samples. Recently a new type of cell had been ntroduced [2]. Ths coaxal-crcular cell uses cylndrcally shaped samples and avods the dsadvantages of the common coaxal cell. It conssts of two coaxal wavegudes connected through to a central cylndrcal secton. In other words, the coaxal wavegudes are abruptly truncated at the faces of the sample, whch resdes n a central cylndrcal secton and maes an electrcal contact wth the central electrodes of coaxal wavegudes. The sample has a smple cylndrcal geometry. We adopted ths concept and made a cell for 3.81 cm (1.5 nch) dameter cores. We wll refer to our cell as a cylndrcal cell or a cylndrcal delectrc cell. FORWARD MODEL AND INVERSION METHODOLOGY: We utlzed a full wave model developed by Habashy [2]. The forward model approxmates concal coaxal electrodes as coaxal electrodes of constant radus. The valdty of ths approxmaton was verfed by a numercal modelng. We created a model wth the exact dmensons of the cell and computed the response of a hypothetcal sample. The modeled sample had permttvty of 8 and conductvty of.1 S/m at all frequences. Ths sample was meant to resemble cell response to.1 S/m brne. The comparson between the forward model (assumng coaxal electrodes wth a constant radus) and the numercal modelng wth actual cell dmensons s presented n Fg. 2. The agreement between numercal results and the forward model valdates our modelng of the concal electrodes as f they were of constant radus. Determnaton of permttvty and conductvty of a sample from measured S parameters requres an nverson approach. We utlzed an nverson methodology developed by Habashy [2] and based on a Gauss-Newton mnmzaton algorthm. The cost functon was defned as the dfference between the measured and the predcted responses weghted n proporton to the measurement confdence: M 2 C( x) = W ( S ( x) m ) (1) = 1 M s the number of measurements (n our case there are four complex S parameters and M=8), m s the observed response (measured data) and S s the correspondng j j

3 SCA24-2 3/12 smulated response as predcted by the vector of model parameters, x, x [ x... ] = 1 x N, where N s the number of unnowns (N=2, correspondng to conductvty and delectrc constant). W s the measure of the confdence n m. The nverson had been constraned by a non-lnear transformaton. If x max s an upper bound on the model parameter x and x mn s a lower bound, then n order to ensure that x < x < at all teratons, we ntroduce the followng transformaton, x mn x max x x c + 1 max mn 2 = xmn + c, < < + 2 It s clear that x x mn, as c x, as ± x max j c (2) c (3) A Newton mnmzaton approach s based on a local quadratc model of the cost functon. The quadratc model s formed by tang the frst three terms of the Taylorseres expanson of the cost functon around the current -th teraton ( x ), as follows, T 1 T C( x + p ) C( x ) + g ( x ) p + p G( x ) p, (4) 2 where the subscrpt T ndcates transposton and p = x +1 x s the step n x towards the mnmum of the cost functon C (x). The vector g( x) = C( x) s the gradent vector of the cost functon C(x) and G( x) = C( x) s the Hessan of the cost functon C (x). For the reconstructon of the conductvty and permttvty we utlzed a varaton of the Newton approach nown as the Gauss-Newton algorthm. In the Gauss-Newton method, one dscards the second order dervatves to avod the expensve computaton. The Gauss-Newton mnmzaton approach has a rate of convergence that s slghtly less than quadratc but sgnfcantly better than lnear. It provdes quadratc convergence n the neghborhood of the mnmum. The teraton process stops f the dfference between two successve terates, ( +1) -th and -th, of the model parameters are wthn a prescrbed tolerance factor, tole, of the cost functon at the current terate: C ( x + 1 ) C( x ) tole C( x + 1) (5) The completon of the nverson process s acheved at cost functon mnmum. There are only two model parameters (sample permttvty and conductvty). It allows for a graphcal representaton of the cost functon dependence on these parameters. In Fg. 3 the cost functon magntude at 1 MHz s plotted aganst the vertcal axs. The permttvty and conductvty span n the horzontal plane. The map was calculated for a reflecton measurement. True sample permttvty s 5 and conductvty s.3 S/m.

4 SCA24-2 4/12 There s a sngle mnmum of the cost functon at ε =5 and σ =.3 S/m. Hence at 1 MHz the choce of an ntal vector of model parameters (ntal guess) should not nfluence the result of nverson. The cost functon behavor dffers sgnfcantly at hgh frequences. Fg. 4 dsplays analogous map at 2 GHz. Besdes the correct mnmum at ε =5 and σ =.3 S/m there are several false mnma. If the ntal guess of the model parameters has been chosen suffcently far from the true values of permttvty and conductvty the nverson process wll lely be trapped n one of the false mnma. In order to safeguard aganst ths problem we utlze the nowledge of conductvty and permttvty at lower frequences. For every next ( +1) -th measurement we set the ntal guess to the values acheved at the prevous (lower frequency) -th step. Delectrc permttvty and conductvty are contnuous functons of frequency and wth suffcently close spacng between subsequent measurements ths algorthm provdes a relable reconstructon of model parameters. EXPERIMENTAL SETUP AND PROCEDURE A photograph of the expermental apparatus s shown n Fg. 1. The coaxal-crcular cell s connected wth a standard 7 mm coaxal cable to an Aglent 8753 ES Networ Analyzer. The cell s made of stanless steel. The sample s located n a crcular secton sandwched between two tapered coaxal electrodes. The coaxal electrodes are tapered so as to act as an adapter between the networ analyzer's coaxal cable and the crcular sample holder of the cell. The sample s n electrcal contact wth the center conductors of the coaxal sectons of the cell whch press aganst the sample's two flat ends. The dameter of the sample holder s 3.81 cm, the center conductor of the coaxal secton has a dameter of 1.17 cm. The space between the center and outer conductors of the coaxal secton s flled wth Teflon. The characterstc mpedance of the coaxal secton s 5 Ohms. The cell was desgned to mae hgher order TM modes reflected from the coaxalcrcular junctons evanescent. The coaxal sectons are suffcently long to suppress these modes up to the hghest frequency n our experments. The scatterng parameters (S-parameters) across the cell termnals are measured by the networ analyzer. The data acquston s automated and controlled by a LabVew applcaton. All four S-parameters are recorded n the form of an ampltude (n db) and phase (n degrees). The networ analyzer s calbrated wth a conventonal full two-port calbraton set. Ths set ncludes a short, an open and a load standards. Ths calbraton establshes the plane of measurements at the connecton between the analyzer's coaxal cables and the cell termnals. Although the conventonal calbraton s effectve n removng systematc analyzer and cable errors, mperfectons n manufacturng process affect the cell response; the concal end peces may not respond exactly le 5 Ohm coaxal lne. In order to correct for these mperfectons, several addtonal procedures have been utlzed. One possblty s to move the plane of calbraton to the faces of the cell s coaxal elements, whch could be acheved by calbratng the networ analyzer at the end of the cell coaxal sectons. Ths

5 SCA24-2 5/12 approach has the advantage of provdng complete error correcton but requres new calbraton standards to be made; a standard Hewlett-Pacard calbraton set cannot be connected to the open face of the coaxal secton. We found t dffcult to mae standards wth the requred response due to varous reasons ncludng non-reproducble connectons and eventually abandoned the effort. Another way to enhance the accuracy of measurements s to remove the nfluence of the cones after the measurement. Ths could be done f the S matrx of each cone s nown. The S matrx conssts of four complex numbers descrbng an object s ablty to transmt and reflect electromagnetc waves. Due to the symmetry of ths matrx one has to defne three ndependent complex scatterng parameters. The lmtng factor for ths correcton s the accuracy to whch the S matrx can be determned. We have made three reflecton measurements from dfferent objects wth nown reflecton response for each concal coaxal end pece. These three measurements can provde suffcent nformaton to determne the S matrx when the reflectng objects have suffcently dfferent responses. An deal set would nclude an open, a short and a perfect termnaton (5 Ohm load) but that set would leads us bac to the problem of manufacturng good calbraton standards for a non-standard connecton. An alternatve method that avods the connecton problem s to use lquds wth nown propertes as reflecton standards. We could not dentfy three lquds wth suffcently dfferent scatterng propertes that a relable S matrx determnaton could be made for the entre frequency range. In the end, we used a smple normalzaton. A correcton based on a sngle measurement of a nown standard has been successfully used prevously [2]. In that case the data was normalzed wth the measurement from an empty cell. We found we get more relable results when the measurement s normalzed to a materal of unform propertes, whch are smlar to those of the sample to be tested. We also assessed the uncertanty assocated wth the measurement. Sample permttvty and conductvty can be estmated from each of the four measured complex S-parameters. The nverson of these four values can yeld four slghtly dfferent pars of conductvty and permttvty values due to the nhomogenety of the roc sample. The goal s to combne these values n such a way that the uncertanty wll be mnmzed. The accuracy of the ampltude and the phase measurements s lmted to the accuracy of the networ analyzer. For our expermental set up we estmated that the ampltude s nown to wthn ±. 3 db, and the phase angle wthn ±.5. Let subscrpt denote values obtaned from nverson of -th S-parameter. These values consst of the real part of delectrc constant ε ' and the complex part ε ". We compute the change n the delectrc constant alternately varyng the phase and the ampltude of the -th S- parameter wthn the accuracy of the measurement [1]. Corresponded change n the value of ε' s mared as ε' j (j=1,2 for permttvty and conductvty).

6 SCA24-2 6/12 Let 2 2 w' = 1 ( ( ε ' ) ) (6) j = 1 j The combned estmate of ε ' from four S-parameters s then obtaned by tang a weghted average of ε wth the weght factor w' ε ' w' (7) = w ' ε ' / The uncertanty ε ' assocated wth the ε ' s found usng the expresson 1/ 2 ε ' = w ' w' (8) A smlar procedure s appled to the magnary part ε ". RESULTS The accuracy of our measurement has been verfed on fluds wth nown delectrc propertes: brnes and glycerol. Brne delectrc constant and conductvty n a wde frequency band can be predcted wth the Klen-Swft model. The requred nputs are the DC brne conductvty and temperature. Comparson between expermentally obtaned permttvty and conductvty of 1 Ohmm brne and Klen-Swft model predcton s shown n Fg. 5. Error bars are suppressed for clarty. Generally there s a good agreement between the measurement and the model. Around 1 GHz there s a slght dscrepancy between nverted and modeled results. These dscrepances correspond to the propagaton of the low order modes through the cylndrcal wavegude. For olfeld rocs the delectrc constant does not reach suffcently hgh values around 1 GHz to allow mode propagaton and the measurement results are not affected. The delectrc constant of brnes doe not change sgnfcantly n the expermental frequency range. Also at 1 GHz, brnes exhbt notably hgher permttvty than saturated cores. Another set of measurements was made on pure organc lquds such as glycerol and 1,3 propanedol. Glycerol permttvty dsperson resembles the behavor of brnesaturated rocs. The measured permttvty and conductvty are shown n Fg. 6. The obtaned values are close to the ones reported n [4]. The uncertanty n the permttvty measurement s gettng larger wth decreasng frequency. Ths happens due to the fact that the sample sze s becomng smaller compare to a wavelength causng the decrease n senstvty to the sample characterstcs. The magntude of the uncertanty at a gven frequency s mostly dependent on the sample conductvty and ncreases wth ncreasng conductvty. Measurement on a cylndrcal lmestone core fully saturated wth ol s shown on the Fg. 7. Ths sample has 29% porosty. As t s expected there s no frequency dependence of permttvty. If the porosty s nown then the matrx permttvty can be calculated from the measurement of a dry (or ol saturated) sample. Ths s an mportant capablty for geophyscal applcatons.

7 SCA24-2 7/12 Fnally, the results of our measurements on partally saturated lmestone are presented on the Fg. 9. The delectrc permttvty and conductvty of the roc decreases wth decreasng brne saturaton. The ablty to mae measurements at partal saturatons allow for a comparatve analyss of mxng laws at selected frequences. Such an analyss for carbonate rocs has been carred out [9]. CONCLUSION We have descrbed a cylndrcal cell for a measurement of delectrc permttvty and conductvty over a wde frequency range. The cell has a concally shaped coaxal transton between networ analyzer cables and coaxal-crcular juncton at the face of the sample. The concal end peces are necessary n order to accommodate samples larger than the dameter of cable connectons. Good agreement between the forward model predctons and the numercal modelng calculatons valdates the forward model and justfes the treatment of the concal-coaxal sectons as coaxal cylnders, whch were ncorporated n the forward model. The delectrc permttvty and conductvty of the sample are obtaned from an nverson of measured S-parameters. The nverson s based on a Gauss-Newton mnmzaton wth a varable ntal guess. Ths methodology allows for a fast and relable reconstructon of the permttvty and conductvty from the measured data. Measurement uncertantes have been estmated and error bars are computed based on senstvty of complex delectrc constant to the varaton n S-parameters. Fnally, we have establshed good agreement between measured and expected values of delectrc permttvty and conductvty for 1 Ohmm brne. RF propertes of the salt soluton were derved from the DC measurements of temperature and conductvty based on Klen-Swft model. ACKNOWLEDGEMENT The authors would le to than Mattheu Smon for carryng out numercal smulatons, Jean-Baptst Clavaud for developng a Labvew applcaton, Abby Matteson, Wave Smth and Phllp Frulla for preparng roc samples. REFERENCES 1. Calvert, T.J., Rau N.R., Electromagnetc propagaton A new dmenson n loggng, SPE Habashy, T., Taheran R., Yuen J., Kong J., A coaxal-crcular wavegude for delectrc measurement, IEEE Trans. Geoscence and Remote Sensng, vol.29, No.2,

8 SCA24-2 8/12 3. Klen, L., and Swft, T., An mproved model for the delectrc constant of sea water at mcrowave frequences, IEEE Trans. On Antennas and Propagaton, vol. AP-25, No.1, Bucley F., and Maryott A., Tables of delectrc dsperson: data for pure lquds and dlute solutons, Natonal Bureau of Standards Crcular 589, November 1, H.E. Bussey, Measurement of RF propertes of materals. A survey, Proc. IEEE, vol.55, pp , L.C. Shen, A laboratory technque for measurng delectrc propertes of core samples at ultra hgh frequences, SPE 12552, W.B. Wer, Automatc measurement of complex delectrc constant and permeablty at mcrowave frequences, Proc, IEEE, vol. 62, no.1, pp.33-36, R.N. Rau. And R.P. Wharton, Measurement of core electrcal parameters at UHF and mcrowave frequences, SPE 938, 55 th annual meetng of the SPE, Dallas, Texas, Sept , Seleznev, N., Boyd, A., Habashy, T., Luth, S., Delectrc mxng laws for fully and partally saturated carbonate rocs, Proceedngs of the SPWLA 45th Annual Loggng Symposum, June 6 9, 24, The Netherlands 1. Banavar, J.R., Sen, P.N., Tomanc, J., Wong, P., Inverson method for obtanng delectrc data from scatterng parameters, Schlumberger nternal report. Fg. 1. Cylndrcal delectrc cell connected to a networ analyzer.

9 SCA24-2 9/12 2 Reflecton phase 5 Reflecton ampltude Tra nsmsson phase Transmsson ampltude Fg.2. Comparson between numercal smulatons of the cell response (dots) and the forward model (sold lnes). Sample permttvty s 8 and conductvty s.1 S/m. Fg. 3. Cost functon map at 1 MHz

10 SCA24-2 1/12 Fg. 4 Cost functon map at 2 GHz 1.1 S/m brne Permttvty Conductvty S/m Fg. 5. Measured (blac) permttvty and conductvty of a.1 S/m brne versus Klen-Swft model (red).

11 SCA /12 1 Glycerol 8 Permttvty Conductvty S/m Permttvty Fg. 6. Permttvty and conductvty of a pure glycerol Ol saturated lmestone Conductvty S/m Fg. 7. Lmestone fully saturated wth ol. Porosty s 29 pu.

12 SCA /12 5 Indana Lmestone 45 4 Permttvty Fg. 8. Lmestone at four dfferent partal saturatons. Water saturaton decreases wth decreasng delectrc constant.